Regularization Methods For Ill-posed Problems Based On Perturbation Of Fractional Power

It is known to us that there are many physical systems evolving in time which are ill-posed. Such problems can be reduced to an ill-posed Cauchy problem. The aim of study ill-posed Cauchy problem is to give a theoretical instruction and methods for the practical ill-posed problems. In this paper, we are devoted to study the ill-posed abstract Cauchy problem.Because of the ill-posedness of the problem, the normal method of finding the approx-imate solution directly is vain. To solve such ill-posed problem, we need the regularization methods. There are already a variety of regularization methods existed. One of the widely used method is Quasi-reversibility method. The main idea of the method is:adding an ap-propriate perturbation to the original problem, the solution of the ill-posed problem can be approximated by the solution of the well-posed problem, therefore we obtain the approxi-mate solution indirectly.In this paper, we improve two regularization methods by adding a perturbation of frac-tional powers.In Chapter3, we improve the stochastic regularization method introduced by Dalecky and Goncharuk. The perturbation in the original regularization is replaced by fractional powers, then we get a new regularization method which includes the original stochastic regularization method as a particular case. And there are three improvements as follow:1. One of the main improvements is the fact that the restriction of the angle of the homomorphic semigroup generated by-A is relaxed from (π/4, π/2] to (0,π/2]. Then Our result is more convenient in application to differential operators.2. Since the exponent of the fractional power which perturbing the ill-posed problem can be chosen in an interval, for different exponents, there are differential families of regular-izing operators. Therefore, in the application, we can find different families of regularizing operators via choosing different exponents according to the practical condition.3. Another improvement appears in the estimate of the solution operator of the approx-imate equation. We obtain a better one which illustrates clearly the asymptotic behavior of the solution and also the relationship between the solution and the regularization parameter.In Chapter4, we improve the Gajewski-Zacharias Quasi-reversibility method. The perturbation in the original regularization is replaced by fractional power, then we get a new regularization method which includes the original regularization method as a particular case. And there are two improvements as follow:1. Since the exponent of the fractional power which perturbing the ill-posed problem can be chosen in an interval, for different exponents, there are differential families of regular-izing operators. Therefore, in the application, we can find different families of regularizing operators via choosing different exponents according to the practical condition.2. The regularizing operators in the modified method are smaller than those in original method, therefore the new method is more convenient in the practical application.